The development of high-intensity LED bulbs and their improved energy efficiency per unit of illumination intensity has led to the development of LED fixtures for street lighting, parking lot lighting, interior lighting and other applications in which conventional sodium vapor, mercury vapor, fluorescent or incandescent lights were and still are being used. A problem with LED fixtures arises from the fact that a plurality of LED bulbs is required to provide illumination equivalent to a single mercury vapor, sodium vapor or incandescent bulb. LED bulbs and lens members for LED bulbs generally divert the light from the LED into a circular pattern. In order to illuminate a large area with LEDs, it is generally necessary to use a light fixture on which a plurality of LEDs are arranged in an array with corresponding lens members that direct the light from the LEDs into overlapping circular beams. Due to the high amount of overlapping area needed to cover an area that is to be illuminated with light beams projecting a circular pattern, use of LED fixtures has resulted in non-uniform illumination, including a patterned distribution of bright, over-illuminated areas and relatively darker, less illuminated areas. Additionally, although LEDs are energy efficient, the number of LEDs needed to provide a desired level of illumination to a given area from a lighting fixture can be reduced if the overlapping areas can be reduced or eliminated. This would reduce both the cost of manufacturing the fixture and the energy cost of operating the fixture.